A number-theoretic estimate for the thomas-fermi density

In this paper we obtain an estimate for the Thomas-Fermi density which plays a role in the analysis of the atomic enegry asymptotics. Such estimate has obvious number-theoretic features related to the radial symmetry of a certain Schrodinger operator, and we use number-theoretic methods in our proof. From the technical viewpoint, we also simplify and improve some of the original estimates in the proof of the Dirac-Schwinger correction to the atomic enegry asymptotics.